Tag Archives: Math

This Year in Uranium Decay

Pumice from the Bishop Tuff (~767 ka).  Zircons in this pumice are rich (relatively) in uranium, with up to 0.5% U.[1,2]  Image credit: Bill Mitchell (CC-BY).
Pumice from the Bishop Tuff (~767 ka). Zircons in this pumice are rich (relatively) in uranium, with up to 0.5% U.[1,2] Image credit: Bill Mitchell (CC-BY).

With 2016 now upon us, I felt it would be appropriate to think about what a new year means for uranium geochronology. What can we expect from the year ahead? Without getting into any of the active research going on, I felt it would be useful to address simply what is physically happening.

On Earth, there is roughly 1×1017 kg of uranium.[3] The ratio of 238U:235U is about 137.8:1, and 238U has a mass of roughly 238 g/mol (=0.238 kg/mol). Looking only at 238U, that gives us
1x1017[kg]x(137.8/138.8)/0.238[kg/mol] = 4.17x1017 mol [238U]

Radioactive decay is exponential, with the surviving proportion given by e-λt where λ is the decay constant (in units of 1/time) and t is time, or alternatively, e-ln(2)/T1/2*t, where T1/2 is the half-life and t is time.

To find the proportion that decays, we subtract the surviving proportion from 1: (1-e-λt)

Multiplying this proportion by the number of moles of 238U will give us the moles of decay, and multiplying by the molar mass will give the mass lost to decay:

(1-e-λt)*molU

Plugging in numbers, with λ238 = 1.54*10-10 y-1, t = 1 y and the moles of 238U from above, we get:

(1-e-1.54*10-10)*4.17*1017 mol [238U] = 6.4*107 mol

That yields (with proper use of metric prefixes) roughly 64 Mmol U decay, or 15 Gg of U on Earth that will decay over the next year.

Although those numbers sound very large, they are much smaller than even the increase in US CO2 emissions from 2013 to 2014 (50 Tg, or 50,000 Gg); total US CO2 emissions in 2014 were estimated at 5.4 Pg (=5.4 million Gg).[US EIA]

As for what’s in store for geochronology as a field, I think there will be a lot of discussion and consideration regarding yet another analysis of the Bishop Tuff.[4] Dating samples which are <1 Ma (refresher on geologic time and conventions) using U/Pb can be tricky, and Ickert et al. get into some of the issues when trying to get extremely high-precision dates from zircons. The paper is not open access, but the authors can be contacted for a copy (@cwmagee and @srmulcahy are active on Twitter, too!).

***
[1] J. L. Crowley, B. Schoene, S. A. Bowring. “U-Pb dating of zircon in the Bishop Tuff at the millennial scale” Geology 2007, 35, p. 1123-1126. DOI: 10.1130/G24017A.1
[2] K. J. Chamberlain, C. J. N. Wilson, J. L. Wooden, B. L. A. Charlier, T. R. Ireland. “New Perspectives on the Bishop Tuff from Zircon Textures, Ages, and Trace Elements” Journal of Petrology 2014, 55, p. 395-426. DOI: 10.1093/petrology/egt072
[3] G. Fiorentini, M. Lissia, F. Mantovani, R. Vannucci. “Geo-Neutrinos: a short review” Arxiv 2004. arXiv:hep-ph/0409152 and final DOI: 10.1016/j.nuclphysbps.2005.01.087
[4] R. B. Ickert, R. Mundil, C. W. Magee, Jr., S. R. Mulcahy. “The U-Th-Pb systematics of zircon from the Bishop Tuff: A case study in challenges to high-precision Pb/U geochronology at the millennial scale” Geochimica et Cosmochimica Acta 2015, 168, p. 88-110. DOI: 10.1016/j.gca.2015.07.018

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Geoscientist’s Toolkit: LaTeX

A LaTeX file.  Image/text credit: Bill Mitchell.
A LaTeX file. Image/text credit: Bill Mitchell.

LaTeX is a typesetting program used for preparing documents—generally articles and books, but sometimes posters and presentation slides. It is available for Linux, Mac, and Windows, and is free, open-source software. The primary output file format these days is PDF, but other options are available.

When you are putting together a document with figures, citations, and sections which get moved around, it is tough to use a common word processing program and maintain sanity. However, because LaTeX is a markup language (like HTML, the HyperText Markup Language), it is explicit which text is grouped where. For instance, suppose you are trying to have both superscripts and subscripts following a letter, such as in CO32-. If you need to edit the superscripts or subscripts in Word, it can get confused easily. In LaTeX, it is explicit which parts are superscript and which are subscript. A little more work up front saves a lot of frustration later.

Above is a small excerpt from my dissertation, written in LaTeX. I am not sure I would have survived grad school had I attempted to write my dissertation in a word processor.

Yes, there is a learning curve to using LaTeX, and you don’t see changes in the finished document immediately when you make them in your text editor, but there are tons of advantages.

First, the format is all plain text, so it will be readable for a long time and across platforms (although the OpenDocument formats are attempting to make word processor documents future-compatible). Plain text is also very convenient when combined with things like version control software. Track changes isn’t just for word processors!

LaTeX separates the content from the formatting. Most of the formatting is done automatically. Yes, you manually specify that something is a emphasized, or is part of a quote, or a heading, but LaTeX will make sure that the formatting is consistent throughout (unless you intervene), and the defaults are generally good.

One place where LaTeX really shines is in mathematical equations. Greek letters and many mathematical symbols are input as commands such as \beta or \sqrt{n}, so your hands need never leave the keyboard. Once typeset, the equations are neat and properly sized.

Many journals accept submissions in LaTeX, if they do not outright encourage its use, because it is easy to keep the formatting consistent from article to article. The fonts will match, the font sizes will match, and in general things are awesome and look professional.

I have given several presentations made with LaTeX (pdf output). The outline slides are automatically maintained, and slide headers/footers can show where in the presentation you are. Those indicators link to the sections if you click on the section name, and it’s all done automatically. LaTeX is totally worth the effort of learning, and do it soon while you can take your time and experiment. Writing your dissertation while learning LaTeX is a recipe for unhappiness.

So, now that you’re ready to get started, here are some tutorials and reference materials:

It really bothers me when the justification for doing something slow, inefficient, and expensive is “that’s what most people use, and I can’t be bothered to learn something new.” There comes a time to do things differently, and a good ecosystem is one where there are several options based around open standards. Case in point: USB ports are great! The proprietary charger connection on my (old-school) phone? Awful. Lock-in is expensive. Choose open source.

Space and Gravity

Solar flare, with Earth for scale.  Image credit: Karl Battams and NASA SDO.
Solar flare, with Earth for scale. Image credit: Karl Battams and NASA SDO.

Space: An Out-of-Gravity Experience opened last week at the Science Museum of Minnesota. I have seen the exhibit, and it is spectacular.

One of the major points the exhibit makes is about gravity, and whether or not there is gravity at the International Space Station, or on the way to Mars or other planets.

Here on Earth, the influence of gravity is pretty obvious. If you throw a ball, it will fall back to Earth fairly quickly. Were you on the Moon or Mars, you could hop around in your space suit, secure in the knowledge that gravity would pull you back down, and insecure about whether your rocket would take you back to Earth.

I will answer that with two questions, which you could imagine asking a stranger in this order:

  1. Why are astronauts are apparently weightless when here on Earth we’re firmly held?
  2. Why, despite the Sun’s great mass, do we not fall into the Sun?

Why don’t we fall into the Sun?! What an absurd question to think about!

The answer has to do with what it means to be in orbit. Randall Munroe of XKCD explains:*

Space is like this:

Imagine you have a machine which can shoot a baseball horizontally at any speed you choose (under the speed of light, and you probably don’t want it getting near that fast anyway). As with many physics thought experiments, we will ignore air resistance. When you start out at reasonable speeds, similar to that which a baseball pitcher throws, it will hit the ground. As you increase the speed, it goes farther and farther before reaching the ground. Keep this in mind.

The Earth is not flat. Although it can seem that way in some areas, the Earth is indeed roughly spherical. At some speed, the baseball will fall toward the Earth in an arc which parallels the Earth’s surface. That speed is the orbital speed (about 8 km/s).** Above about 11 km/s, the baseball would leave the Earth, never to return unless affected by another body. Orbiting the Sun, Earth is moving around 30 km/s along its orbit.

When astronauts experience weightlessness, it is because the spaceship they are in is falling to Earth at the same rate as they are. Think of it like a roller-coaster starting it’s drop, but that just keeps going down and down and down.

Here’s another question for you: which body exerts more gravitational pull on you, the Earth, or the much-more-massive Sun?

To find out, let’s do some (easy!) math. We know from Wikipedia that the attractive force due to gravity is G*m1*m2*d-2, where G is the gravitational constant, m is the mass of the two objects in question (e.g. you and the Earth), and d is the distance between their centers. Approximate a human mass as 60 kg, use your favorite search engine to find the mass of the Earth and the Sun in kg, the radial distance of the Earth in meters, and the orbital distance of the Earth from the Sun in meters, and plug away.

The result? Earth pulls on you more than the Sun, by a factor of ~1600. This makes sense. If the Sun pulled you more strongly than the Earth, you would move toward the Sun, fall in, and die.

Between the Sun and the Earth, there is a point where the gravitational influences and centrifugal force all cancel each other out. This point is the Lagrangian point L1, and is a gravitational sweet spot much like an orbital speed is just right.

Now that things are just right, I should probably wrap things up. Any further addition will send me diving down the rabbit-hole again!

* Incidentally, xkcd and particularly the xkcd what-if comics are among my favorite things to read. I almost think an Ig-Nobel Prize might be deserved for cartoons which make us laugh, then think.

** The orbital speed is somewhat dependent on orbital distance; things further out have a slightly lower orbital speed.